On polyhomogeneous symbols and the Heisenberg pseudodifferential calculus

Nathan Couchet (1) , Robert Yuncken (2)

1 LMBP - Laboratoire de Mathématiques Blaise Pascal
2 IECL - Institut Élie Cartan de Lorraine

Polyhomogeneous symbols, defined by Kohn-Nirenberg [16] and Hörmander [15] in the 60's, play a central role in the symbolic calculus of most pseudodifferential calculi. We prove a simple characterisation of polyhomogeneous functions which avoids the use of asymptotic expansions. Specifically, if U is open subset of R d , then a polyhomogeneous symbol on U × R d is precisely the restriction to t = 1 of a function on U × R d+1 which is homogeneous for the dilations of R d+1 modulo Schwartz class functions. This result holds for arbitrary graded dilations on the vector space R d. As an application, using the generalisation T H M of A. Connes' tangent groupoid for a filtered manifold M , we show that the Heisenberg calculus of Beals and Greiner [3] on a contact manifold or a codimension 1 foliation coincides with the groupoid calculus of [29].

Mots clés: pseudodifferential calculus; classical symbol; homogeneous Schwartz function; tangent groupoid of a Heisenberg manifold; filtered manifold; contact manifold; non-commutative geometry.

Domaines: Géométrie différentielle [math.DG]; Analyse fonctionnelle [math.FA]; Algèbres d'opérateurs [math.OA]

Nathan Couchet, Robert Yuncken. On polyhomogeneous symbols and the Heisenberg pseudodifferential calculus. 2022. ⟨hal-03832219⟩